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It is mathematics that offers the exact mathematical sciences a certain measure of security which, without mathematics, they could not obtain.
Eratosthenes and the Mystery of the Stades
Parallel light rays
2. That light rays from the Sun which strike the Earth are parallel.
In fact, this assumption is incorrect. Sunrays striking the Earth are not parallel. How did Eratosthenes justify such a claim? Just years earlier a man named Aristarchus of Samos (310-230 BCE) produced a work entitled On the Sizes and Distances of the Sun and Moon. This masterpiece of ancient astronomy contains an elaborate geometric proof which asserts that the distance from the Earth to the Sun is approximately equal to 180 Earth diameters. Furthermore, he reasoned that the Sun’s diameter is approximately 6 3/4 times that of the Earth. Actually, the Sun is almost 1200 Earth diameters from the Earth, and the Sun’s diameter is around 109 times the Earth’s, but the idea is the same – the Sun is much larger than the Earth, and light rays from the Sun travel a great distance to the Earth [8, pp.350-352 ].
The shaded regions represent the difference between the assumed parallel sunrays and actual nonparallel sunrays. Using Aristarchus’ measurements and some modern mathematics, we can judge the significance of this difference. Consider one of the shaded regions.
Notice that the shaded region is a right triangle. Angle b, at the farthest vertex of the triangle, is the angular difference between the actual sunrays and the assumed parallel sunrays. Using the information provided by Aristarchus, angle b can be approximated. According to Aristarchus, the distance to the Sun is equal to 180 Earth diameters. So the length of the shaded triangle is 180 Earth diameters. Aristarchus also tells us that the Sun’s diameter is equal to 6¾ Earth diameters. Subtracting one Earth diameter from the center of the Sun’s diameter gives us 5 3/4. Dividing by 2, we find that each shaded triangle has a height of (1/2)(5 3/4) = 23/8 Earth diameters.
Using modern trigonometry, we get Tan b = [(23/8) Earth diameters]/ 180 Earth diameters; so b = Tan-1(23/1440), or b is approximately 0.915 degrees.
At the time of Aristarchus and Eratosthenes, the instruments used to make angular measurements were so crude that an error of less than a degree was negligible [4, p.57]. Of course, Aristarchus and Eratosthenes did not have the benefit of our modern trigonometry, but using the Euclidean geometry available to them they were able to recognize that the small angular difference was relatively insignificant [5, p.154 ]. With this idea in mind, Eratosthenes is justified in making the assumption that sunrays striking the Earth are parallel.